Find dy/dx y = square root of 3-x

Solution:

Step:1

Convert $\sqrt{3 - x}$ to exponential form using the rule $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$.

$$y = (3 - x)^{\frac{1}{2}}$$

Step:2

Take the derivative of both sides with respect to $x$.

$$\frac{dy}{dx} = \frac{d}{dx}((3 - x)^{\frac{1}{2}})$$

Step:3

The derivative of $y$ with respect to $x$ is $\frac{dy}{dx}$.

$$\frac{dy}{dx}$$

Step:4

Differentiate the right-hand side of the equation.

Step:4.1

Apply the chain rule for differentiation, $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$, where $f(x) = x^{\frac{1}{2}}$ and $g(x) = 3 - x$.

Step:4.1.1

Set $u = 3 - x$ and differentiate.

$$\frac{d}{du}(u^{\frac{1}{2}}) \cdot \frac{d}{dx}(3 - x)$$

Step:4.1.2

Use the power rule $\frac{d}{du}(u^n) = nu^{n-1}$, where $n = \frac{1}{2}$.

$$\frac{1}{2} u^{\frac{1}{2} - 1} \cdot \frac{d}{dx}(3 - x)$$

Step:4.1.3

Substitute $u$ back in as $3 - x$.

$$\frac{1}{2} (3 - x)^{\frac{1}{2} - 1} \cdot \frac{d}{dx}(3 - x)$$

Step:4.2

Express $-1$ as a fraction with a common denominator by multiplying by $\frac{2}{2}$.

$$\frac{1}{2} (3 - x)^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \cdot \frac{d}{dx}(3 - x)$$

Step:4.3

Combine $-1$ and $\frac{2}{2}$.

$$\frac{1}{2} (3 - x)^{\frac{1}{2} + \frac{-1 \cdot 2}{2}} \cdot \frac{d}{dx}(3 - x)$$

Step:4.4

Simplify the exponent by combining the numerators over the common denominator.

$$\frac{1}{2} (3 - x)^{\frac{1 - 2}{2}} \cdot \frac{d}{dx}(3 - x)$$

Step:4.5

Simplify the numerator.

Step:4.5.1

Multiply $-1$ by $2$.

$$\frac{1}{2} (3 - x)^{\frac{1 - 2}{2}} \cdot \frac{d}{dx}(3 - x)$$

Step:4.5.2

Subtract $2$ from $1$.

$$\frac{1}{2} (3 - x)^{\frac{-1}{2}} \cdot \frac{d}{dx}(3 - x)$$

Step:4.6

Combine the fractions.

Step:4.6.1

Move the negative exponent to the denominator using the rule $b^{-n} = \frac{1}{b^n}$.

$$\frac{1}{2(3 - x)^{\frac{1}{2}}} \cdot \frac{d}{dx}(3 - x)$$

Step:4.7

Apply the Sum Rule for differentiation to $3 - x$.

$$\frac{1}{2(3 - x)^{\frac{1}{2}}} \left(\frac{d}{dx}(3) + \frac{d}{dx}(-x)\right)$$

Step:4.8

The derivative of a constant is zero.

$$\frac{1}{2(3 - x)^{\frac{1}{2}}} (0 + \frac{d}{dx}(-x))$$

Step:4.9

Add $0$ and the derivative of $-x$.

$$\frac{1}{2(3 - x)^{\frac{1}{2}}} \cdot \frac{d}{dx}(-x)$$

Step:4.10

The derivative of $-x$ is $-1$.

$$\frac{1}{2(3 - x)^{\frac{1}{2}}} \cdot (-1)$$

Step:4.11

Simplify the expression.

$$-\frac{1}{2(3 - x)^{\frac{1}{2}}}$$

Step:5

Combine the left and right sides of the equation.

$$\frac{dy}{dx} = -\frac{1}{2(3 - x)^{\frac{1}{2}}}$$

Step:6

Finalize the derivative.

$$\frac{dy}{dx} = -\frac{1}{2(3 - x)^{\frac{1}{2}}}$$

Knowledge Notes:

1. Exponential Form: The square root of a number can be expressed in exponential form as $\sqrt[n]{a} = a^{\frac{1}{n}}$ where $n$ is the root and $a$ is the base.

2. Derivative: The derivative of a function measures the rate at which the function value changes as its input changes.

3. Chain Rule: A fundamental rule in calculus used to differentiate compositions of functions. It states that $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$.

4. Power Rule: A basic differentiation rule that says if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

5. Sum Rule: The derivative of a sum of functions is the sum of the derivatives, $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$.

6. Negative Exponent Rule: For any nonzero number $b$ and integer $n$, $b^{-n} = \frac{1}{b^n}$.

7. Differentiating Constants: The derivative of a constant is zero. If $c$ is a constant, then $\frac{d}{dx}(c) = 0$.

8. Combining Fractions: When combining fractions with common denominators, you can add or subtract the numerators and keep the common denominator.